// Make newform 8550.2.a.u in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_8550_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_8550_2_a_u();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_8550_2_a_u();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_8550_a();" function MakeCharacter_8550_a() N := 8550; order := 1; char_gens := [1901, 1027, 1351]; v := [1, 1, 1]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; function MakeCharacter_8550_a_Hecke(Kf) return MakeCharacter_8550_a(); end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 2; raw_aps := [[1], [0], [0], [-3], [-2], [1], [3], [-1], [-1], [5], [-8], [2], [8], [-4], [8], [-1], [-15], [2], [-3], [-2], [-9], [-10], [-6], [0], [2], [-2], [6], [-7], [-15], [14], [-18], [-12], [-17], [0], [0], [2], [2], [16], [-12], [-6], [0], [22], [-7], [6], [8], [-25], [27], [-14], [-17], [-10], [-6], [-15], [-8], [-2], [8], [24], [-30], [7], [-28], [8], [6], [9], [12], [-7], [-29], [-27], [17], [32], [-2], [10], [9], [15], [-28], [-29], [15], [-26], [30], [-8], [8], [-20], [0], [-13], [18], [-14], [20], [-26], [-10], [7], [28], [-4], [-2], [20], [2], [28], [40], [39], [30], [28], [-29], [2], [-28], [28], [-36], [-40], [-28], [37], [-12], [34], [0], [-8], [22], [-34], [18], [10], [32], [-42], [26], [23], [-36], [-5], [-23], [-44], [13], [4], [42], [28], [-30], [5], [17], [36], [-40], [-16], [32], [2], [-27], [-35], [9], [17], [3], [15], [-3], [-12], [-29], [23], [-15], [-20], [6], [-12], [-50], [54], [-13], [18], [-34], [-2], [-53], [-12], [5], [55], [7], [-7], [-12], [-46], [-48], [28], [8], [-6], [-8], [-28], [40], [-36], [15], [47], [48], [1], [0], [-30], [-28], [18], [-4], [-40], [47], [23], [26], [28], [39], [30], [7], [56], [-10], [33], [-34], [44], [-8], [-57], [-32], [24], [17], [36], [-37], [9], [30], [-68], [17], [20], [-50], [8], [35], [39], [-45], [-13], [-53], [28], [36], [-27], [0], [-18], [-3], [-42], [58], [-6], [42], [50], [-50], [1], [53], [30], [-46], [30], [-38], [18], [6], [35], [-48], [38], [-14], [-27], [30], [-16], [-50], [-12], [4], [22], [11], [-25], [-36], [-60], [32], [28], [20], [-21], [2], [13], [-37], [50], [44], [-10], [-18], [52], [18], [2], [-64], [-12], [0], [21], [18], [-65], [15], [-27], [-29], [-6], [-33], [47], [-9], [65], [82], [-9], [-42], [55], [-8], [-12], [54], [67], [63], [27], [32], [48], [-9], [-42], [25], [-50], [18], [-12], [54], [28], [-14], [-80], [2], [-46], [35], [42], [26], [53], [-5], [59], [12], [82], [-57], [5], [0], [16], [64], [30], [-82], [36], [-72], [0], [-30], [38], [61], [50], [37], [57], [-27], [-14], [39], [-78], [-65], [-49], [-32], [44], [52], [-27], [35], [34], [42], [68], [45], [-36], [-73], [-73], [11], [-57], [-75], [-58], [-1], [20], [-18], [27], [-2], [48], [-8], [-38], [-32], [36], [-10], [-56], [-35], [-12], [-17], [-66], [-88], [58], [78], [-95], [-38], [-44], [18], [6], [-38], [-42], [45], [59], [5], [-28], [-8], [-5], [-32], [-14], [75], [12], [18], [74], [-13], [78], [-45], [74], [-83], [32], [26], [-32], [50], [29], [-30], [-58], [-12], [-74], [-50], [-20], [-68], [-72], [50], [-86], [-18], [-32], [-90], [-73], [2], [28], [71], [-20], [-4], [18], [19], [2], [-13], [48], [-90], [-8], [3], [-56], [90], [7], [-67], [35], [86], [93], [-61], [5], [-88], [25], [62], [58], [10], [-16], [-58], [-97], [-70], [-38], [52], [-80], [54], [75], [50], [80], [-53], [78], [-4], [-42], [20], [82], [-83], [-12], [-16], [-10], [-78], [78], [-110], [-27], [6], [108], [-65], [-8], [-45], [-23], [-88], [-29], [-20], [-36], [-35], [82], [-4], [73], [40], [-113], [68], [86], [80], [17], [-77], [-61], [-64], [40], [72], [18], [-9], [58], [-40], [-87], [-100], [42], [62], [68], [40], [54], [-80], [92], [52], [-112], [55], [-33], [78], [-79], [-56]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_8550_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_8550_2_a_u();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_8550_2_a_u(:prec:=1) chi := MakeCharacter_8550_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(3593) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_8550_2_a_u();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function MakeNewformModSym_8550_2_a_u( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_8550_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<7,R![3, 1]>,<11,R![2, 1]>,<13,R![-1, 1]>,<17,R![-3, 1]>,<23,R![1, 1]>,<53,R![1, 1]>],Snew); return Vf; end function;